Factorization of polynomials in $\mathbb{Q_p}[X]$

Question

I stumbled upon these two questions while reading Milne's notes on Algebraic Number Theory.

1. Milne's problem 7-6: Let $\gamma=\sqrt{p_1}+\cdots+\sqrt{p_n}$, $p_i$ are distinct primes. We could prove that the minimal polynomial $f(X)$ of $\gamma$ over $\mathbb{Q}$ has degree $2^n$. Now Milne claims that $f$ as a polynomial in $\mathbb{Q}_p[X]$ factors into polynomials of degree at-most $4$ if $p\ne 2$ and degree at-most $8$ if $p=2$. He seems to use the fact that the group $\mathbb{Q}_p^{\times}/\mathbb{Q^{2\times}}$ has size $4$ if $p\ne 2$ and size $8$ if $p=2$. I am unable to connect these two facts.

2. Milne's problem 7-7: Let $\zeta_n$ be a primitive $n$-th root of $1$ for some $n$ relatively prime to $p$, a prime. Milne claims that $[\mathbb{\mathbb{Q}_p}[\zeta_n]:\mathbb{Q_p}]=f$, where $f$ is the order of $n$ modulo $p$. I am again unable to see why this is true. I do know how a prime factorise in cyclotomic extensions.

Is there any general technique that Milne is using to make this claims? More precisely, if we know the minimal polynomial of an algebraic integer over $\mathbb{Q}$, then how does that polynomial factorise in $\mathbb{Q}_p[X]$?

Thanks for your time.

Answer 1

I haven't read Milne's notes, so it's entirely possible there is some other approach that I'm missing here, perhaps using less power, but here it goes...

I'll address the second question first, because I think that's actually the easier one. The degree of the extension $[\mathbb{Q}_p(\zeta_n) : \mathbb{Q}]$ is equal to the product of the ramification and inertia degrees for $p$ in $\mathbb{Q}(\zeta_n) / \mathbb{Q}$ (this should be in Milne's notes, and is certainly in Chapter II of Neukrich). For cyclotomic fields, these are well known - in the case where $p \nmid n$, we have that the ramification degree is 1, and the inertia degree is the order of $p$ modulo $n$ (I believe you have these switched in your statement). So in this case, the degree is the order of $p$ modulo $n$ as claimed, and we don't even need much local field theory.

For the second question, and the best answer that I know to your general question about the factorization of $f(x) \in \mathbb{Z}[x]$ over $\mathbb{Q}_p$, we'll use Proposition II.8.2 from Neukrich. This states:

Suppose the extension $L | K$ is generated by the zero $\alpha$ of the irreducible polynomial $f(X) \in K[X]$. Then the valuations $w_1, \dots, w_r$ extending $v$ to $L$ correspond $1-1$ to the irreducible factors $f_1, \dots, f_r$ in the decomposition $$f(X) = f_1(X)^{m_1} \cdots f_r(X)^{m_r}$$ of $f$ over the completion $K_v$.

Let's specialize to your case. We have that $f(x)$ is a separable polynomial in $\mathbb{Z}[x]$, so it has no repeated roots, even over $\overline{\mathbb{Q}_p}$. We take $K = \mathbb{Q}$, $L = K(\gamma)$, and $v = v_p$. Since $f$ has no repeated roots, all the $m_i$ are equal to $1$, so we get that $f(X) = f_1(X) \cdots f_r(X)$ over $\mathbb{Q}_p$, where the $f_i$ are all distinct, and the number of such polynomials is the number of valuations extending $v_p$ to $L$. I'll note also that the degree of $f_i$ is the degree of the corresponding extension of $K_v$, although this follows only from the discussion surrounding this Proposition in Neukrich, and not the statement itself. The valuations extending $v_p$ to $L$ correspond to extensions of $\mathbb{Q}_p$ that contain $L$. We now need a little Kummer theory to understand these extensions. The fundamental theorem of Kummer Theory tells us the following:

Let $n \geq 1$, and let $F$ be a field containing a primitive $n$th root of unity. Fix an algebraic closure $\overline{F}$ of $F$. There exists a one-to-one, order preserving correspondence between the $n$-Kummer extensions $K / F$ contained in $\overline{F}$, and the finite subgroups of $F^\times / F^{\times n}$.

In your case, the extensions $L_{w_i} / \mathbb{Q}_p$ are all $2$-Kummer, because the Galois group of $L / K$ is $(\mathbb{Z} / 2\mathbb{Z})^r$. Note that this is where we use the assumption that all the $p_i$ are distinct primes (why?). Now you see why the size of $\mathbb{Q}_p^\times / \mathbb{Q}_p^{\times 2}$ matters: the $2$-Kummer extensions of $\mathbb{Q}_p$ correspond to the factors of $f(x)$ over $\mathbb{Q}_p$ by the theorem from Neukrich above, and they correspond to subgroups of $\mathbb{Q}_p^\times / \mathbb{Q}_p^{\times 2}$ by Kummer theory. In Kummer theory the degree of the extension corresponding to a subgroup $A \subset F^\times / F^{\times n}$ is equal to the size of the set, so we see that the extensions of $\mathbb{Q}_p$ that contain $L$ all have degrees bounded by $|\mathbb{Q}_p^\times / \mathbb{Q}_p^{\times 2}|$. Again, these degrees correspond to the degrees of the factors of $f(x)$ over $\mathbb{Q}_p$, and so the statement in Milne follows from the statement about the size of $\mathbb{Q}_p^\times / \mathbb{Q}_p^{\times 2}$.

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References:

1. Neukrich Algebraic Number Theory, Chapter II Section 8 for the parts about valuations and local field extensions;
2. Marcus Number Fields, Theorem 26 for the splitting of $p$ in cyclotomic fields;
3. Guillot A Gentle Course in Local Class Field Theory, Chapter 1 for Kummer Theory.